 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/18/c/ii/solution

Constant translation speed gives

ℓ (t) = v t

, since the initial displacement is zero. Substituting into the Fourier series gives

g (x, t) = \frac{K}{π} n = 1 \sum \infty \frac{( - 1 ) ^{n + 1}}{n} sin [\frac{2 πn}{L} (x - v t)] .

The periodic ramp moves in the positive

x

direction for

v > 0

. At the moving jumps, this series gives the average of the two one-sided values.

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